On a partial converse of the Montessus de Ballore theorem in \({\mathbb{C}}^ n\)
The paper is divided into three sections, namely, {\S}1 Introduction, {\S}2 Notation, definitions and some properties, {\S}3 Convergence. The introduction provides some motivation for the paper, while the next section develops notation, presents definitions and discusses results pertaining to certain algebraic properties. The section on convergence carries the main results of this paper. These are theorems 3.1 and 3.2 which are respectively the Montessus de Ballore theorem and its partial converse. The algebraic properties of the unisolvent rational approximants presented by theorems 2.1, 2.2 and 2.3, are restatements of results due to the author [J. Math. Anal. Appl. 53, 89-98 (1976; Zbl 0319.32005)]. Theorem 3.1, as stated in this paper, is perhaps sharper than what appeared in another paper without proof [Approximation Theory III, Proc. Conf. Hon. G. G. Lorentz, Austin/Tex. 1980, 603-609 (1980; Zbl 0481.41014)]. The new result of the paper is the theorem 3.2. This theorem, even though it is not an exact converse of theorem 3.1, indicates clearly that under less stringent set of conditions, one can still recover the meromorphicity of the function f over \(\Delta^ n\!_{\rho}\).
scientific article; zbMATH DE number 3846139